Part of the reason why each particular label can fit is because there were competing ideologies of the time. The "industrial statesman" or "titan of industry" label could be appropriate because this was the time in American History where much in way of industrial growth, laisssez faire economies, and the business application of Social Darwinism was highly evident. America was in the grip of material advancement and social wealth, and this helped to solidify the belief that the Carnegies, the Stanfords, the Morgans of America of the time were industrial statesmen. Competing with this ideology was the perception of Progressivist ideas which stressed that America cannot be defined by the richest 1%, but rather the economically and socially marginalized 99%. This disproportionate and large percentage of people are the results of the "industrial statesmen." The Progressivists argued that these individuals did not possess economic wealth or material progress, and were subjugated because of it. Due to this, they became the perceived as "robber barons."
The Progressive perception that those who created the wealth stole it from those less fortunate assumes that wealth is a static quantity with a zero-sum gain -- that if Carnegie has it, it must have come at the expense of the hundreds of thousands who didn't. However, this supposition ignores the breakthroughs in technology, banking, and business organization that improved the lives of those hundreds of thousands. Wealth is not zero-sum -- wealth is created, and the individuals who were free to create wealth did so amazingly and benefited the whole of society by raising the standard of living. In some cases, the living standard went from non-existence to bare subsistence -- this describes the effects of rapid industrialization within the Dickensian novels from the 1830's, for example, and describes the US experience with those same issues a generation or two later. The impact of the industrial leader's success is staggering -- between 1850 and 1900, the world population doubled. If very few of those people had the living standard of a Carnegie, it's understandable that at least they existed -- which they would not have without the Industrial Revolution, which would not have occurred without individuals being free to invest their resources as they saw fit.
Nevertheless, where business practices affront individual rights, those engaging in such practices should have been legally required to cease. Poor business practices eventually undermine the business -- who would work for them, or buy their goods or services? If a business evolved into a coercive monopoly, didn't government break it apart to allow other businesses to compete? If a business exploited its workers, didn't government protect them, or did labor unions evolve to address business grievances? Where was this wealth-theft?
The people who ushered in that era of phenomenal growth are creators and innovators.
It is fairly easy to characterize them as robber barons, particularly if you are a progressive reformer of the time. They weren't necessarily against people being rich, they were just against people being obscenely rich, an at the expense of the working poor and immigrants. The top 1% of the population at the time of the late Gilded Age controlled 40% of the wealth in the country, which is a staggering figure.
As industrial statesmen, you could easily argue that they created jobs, fueled the development of a country, and gave back much of their wealth to society in philanthropy (charity). They were the ones who took the risk, invested their money and built their companies, qualities that would qualify them as leaders in any society.
Use this information to judge for yourself which label better fits the wealthy men of the time.
Robber Barons or Industrial Statesmen:
Here is some information that will help you answer the question. I will describe two ways of accumulating wealth.
1) Using bribes and other political influence to get laws and regulations passed so that you can gather in wealth that was created by other people. An example of this is found in the history of railroad building during the years that you are considering. Some railroad builders used bribes and gifts of stock in their railroad ventures to get U. S. Congressmen to pass laws that gave to them (the railroad builders) vast acreages of public lands. They justified this by saying that the country needed the railroads so bad that the government should pay for them by giving the land to the builders. But there were railroad builders who built their roads without these grants of land. James J. Hill is one such. Many big businesses have received subsidies from government, but government has no money of its own; in order to give money to one party, government must first take it from another party. This is the way wealth is gathered from people who created it and given to people who did not create it.
2) Wealth can be accumulated by creating it. For example, if I own a small stand of timber, somebody might pay me $10,000 for it as it is. If I invest $1,000 worth of my time and wear and tear (spelling?) on my chainsaw to cut the trees down and trim the limbs off of them, now somebody might be willing to pay me $12,000 for the same timber. I have just created $1,000 in wealth. (I started with $10,000, and it is now worth $12,000, but I must subtract the $1,000 that I spent, so that my profit is $1,000.) I did this without taking anything from anybody. Government did not give me anything that it had taken from anybody else. I did it all myself.
Which example describes a robber baron and which describes an industrial statesman? Was James J. Hill a robber baron or an industrial statesman?